Abstract:

Both the broken windows only theorem and the bounded image theorem constitute important parts of Thurston’s original proof of the unifomisation theorem for Haken manifolds. According to his plan of proofs in a form of diagram,  his  proof for the bounded image theorem relies essentially on the broken windows only theorem, in particular its second statement. We show that this second statement has counter examples.  Although we can fix it by weakening the result, this weak form cannot be used for the proof of the bounded image theorem.

Nevertheless we can prove that the bounded image theorem itself is valid, making use of recent tools of Kleinian group theorem.

The last part is joint work with Cyril Lecuire (ENS Lyon).

Bio:

Ken' ichi Ohshika, Professor of Gakushuin University and Professor Emeritus of Osaka University

He received PhD from University of Tokyo in 1989, and worked at Tokyo Institute of Technology, University of Tokyo, and Osaka University before joining Gakushuin University. His interest includes Kleinian groups, hyperbolic geometry, Teichmüller theory, and 3-manifolds. He is a recipient of the MSJ geometry prize (2012).

Video: http://archive.ymsc.tsinghua.edu.cn/pacm_lecture?html=Thurston_s_broken_windows_only_theorem_and_bounded_image_theorem.html